Programming Abstractions CS106X Cynthia Lee

Topics Overview Recently: Priority Queue implementations: linked list heap Map interface implementation: Binary Search Tree (BST) While we’re doing trees, another kind of tree: Huffman coding trees (used for compression of files) Today: Hashing: Another Map interface implementation Background: Lookup tables Details: Hashing

Hashing Implementing the Map interface with Hashing/Hash Tables PART 1: Intuition behind the invention of the hash table

Imagine you want to look up your neighbors’ names, based on their house number House numbers: 10565 through 90600 (roughly 1000 houses—there are varying gaps in house numbers between houses) All the houses are on the same street, so we only need to lookup by house number Names: string containing the name(s) living there We will consider two data structure options: linked list, and array of strings Image dedicated to public domain under Creative Commons license: http://commons.wikimedia.org/wiki/File:Salsbury_Row_House.jpg

Option #1: Linked list … Linked list: 10565, “Kyung Suk and Yong Han Lee” 10567, “Isaiah White” 90600, “Josie Spencer and Solange Clark” head … Linked list: Struct has 3 fields: next pointer, int data (house number), and string data (name) Sort them by house number Add/remove: O(n) Find: O(n)

Option #2: Array of strings Index (house number) String value (name) “” 1 … 10565 “Yong Han and Kyung Suk Lee” 10566 10567 “Isaiah White” 90598 90599 90600 “Josie Spencer and Solange Clark” Array of strings: Index is house number, string is name Add/remove: Find:

Array of Strings Array of strings: Array of strings: Index (house number) String value (name) “” 1 … 10565 “Yong Han and Kyung Suk Lee” 10566 10567 “Isaiah White” 90598 90599 90600 “Josie Spencer and Solange Clark” Array of strings: Index is house number, string is name(s)

Array of Strings Array of strings: Array of strings: O(1), O(1) Index (house number) String value (name) “” 1 … 10565 “Yong Han and Kyung Suk Lee” 10566 10567 “Isaiah White” 90598 90599 90600 “Josie Spencer and Solange Clark” Array of strings: Index is house number, string is name(s) Add/remove: ____ Ex.: if somebody moves into the vacant house at 90598, how long would it take to update? Find: ____ Ex.: you want to find the name of the resident at 12475, if any O(1), O(1) O(logn), O(logn) O(n), O(n) Other/none/combination

Array of Strings Array of strings: Index (house number) String value (name) “” 1 … 10565 “Yong Han and Kyung Suk Lee” 10566 10567 “Isaiah White” 90598 90599 90600 “Josie Spencer and Solange Clark” Wow, excellent performance on both!! Only way to do better than O(1) is a time machine that can go back in time and make it take zero/negative time! Everything is awesome (?) Discuss: Can you identify 1-2 specific areas of waste in this approach? Bonus: can you think of a simple fix for at least one of the areas of waste?

One quick fix: Array of strings: Index (house number) String value (name) “” 1 … 10565 “Yong Han and Kyung Suk Lee” 10566 10567 “Isaiah White” 90598 90599 90600 “Josie Spencer and Solange Clark” /* When accessing the array, use array[hash(houseNum)] * rather than array[houseNum] */ int hash(int houseNumber){ return houseNumber-10565; } This solves the problem of the enormous gap from 0 to 10565 So our array size could be ~80,000 entries instead of 90,600 Doesn’t solve the problem of gaps between houses How could we do that? A tricky problem… This approach only works for keys of type int

Hashing Implementing the Map interface with Hashing/Hash Tables PART 2: Getting the MAGICAL performance of our simple house numbers example on any key type, and with less waste

Hash Table is just a modified, more flexible array Keys don’t have to be integers in the range [0-(size-1)] They don’t even have to be integers at all! (Ideally) avoids big gaps like we had with house numbers array Replicates the MAGICAL performance of our array of strings on ANY key/value!! THANK YOU, HASH FUNCTION!! ♥ ♥ ♥ Not necessarily int int in range [0-(size-1)]

hash() function This is where the MAGIC happens! These are typically mathematically sophisticated functions They do their best to ensure a nice uniform distribution of elements across the available array (hash table) They use tricks like modulus (remainder) and prime numbers to do this A lot of art & science, beyond the scope of this class Fun times!

Hashing Implementing the Map interface with Hashing/Hash Tables

Hash table inserts Array index Hashed data 1 2 (A) "Annie", 3 3 (B) "Annie", 3 4 (C) "Annie", 3 5 (D) "Annie", 3 6 7 8 Let’s pretend we have a profoundly not-mathematically-sophisticated hash function: int hash(string key) { return key.length(); } Where does key="Annie" value=3 go? HashMap<string,int> mymap; mymap["Annie"] = 3; See choices in table at right, or: (E) Some other place

Hash table inserts Array index Hashed data 1 2 3 4 5 "Annie", 3 6 7 8 Let’s pretend we have a profoundly not-mathematically-sophisticated hash function: int hash(string key) { return key.length(); } Where does key="Michael", value=5 go? mymap["Michael"] = 5;

Hash table inserts Array index Hashed data 1 2 3 4 5 "Annie", 3 6 7 "Michael", 5 8 Let’s pretend we have a profoundly not-mathematically-sophisticated hash function: int hash(string key) { return key.length(); } Where does key="Michael", value=5 go? mymap["Michael"] = 5;

Hash table inserts Array index Hashed data 1 2 3 4 5 (A) "Annie", 3 7 6 7 (B) "Michael", 5 "Annie", 7 8 Let’s pretend we have a profoundly not-mathematically-sophisticated hash function: int hash(string key) { return key.length(); } Now insert key="Annie", value=7 mymap["Annie"] = 7; See choices in table at right, or: (C) Index 5 should store both "Annie",3 and "Annie",7

Hash table inserts Array index Hashed data 1 2 3 4 5 (A) "Annie", 7 "Maria", 8 6 (B) "Maria", 8 7 "Michael", 5 8 (C) "Maria", 8 Let’s pretend we have a profoundly not-mathematically-sophisticated hash function: int hash(string key) { return key.length(); } Now insert key="Maria", value=8 mymap["Maria"] = 8; See choices in table at right, or: (D) Index 5 should store both "Annie",7 and "Maria",8

Uh-oh! Hash collisions We can NOT overwrite the value the way we would if it really were the same key Can you imagine how you would feel if you used Stanford library HashMap like this and it printed 8?! mymap["Annie"] = 3; mymap["Annie"] = 7; cout << mymap["Annie"] << endl; //expect 7, not 3 mymap["Maria"] = 8; cout << mymap["Annie"] << endl; //expect 7, not 8!!!

Uh-oh! Hash collisions We may need to worry about hash collisions Two keys a, b, a≠b, have the same hash code index (i.e. hash(a) == hash(b)) Need a way of storing multiple values in a given “place” in the hash table, so all user’s data is preserved

Uh-oh! Hash collisions Array index Hashed data 1 2 3 4 5 (A) "Annie", 7 "Maria", 8 6 (B) "Maria", 8 7 "Michael", 5 8 (C) "Maria", 8 There are two main strategies for resolving this: Put the item in the next bin (as in the (B) choice from our previous slide)—this is called “open addressing” Make each bin be the head of a linked list, and elements can chain off each other as long as needed—this is called “closed addressing”

Map Interface: hash-map.h … private: struct node { Key key; Value value; node *next; }; node **buckets; int numBuckets; int count; int hash(const Key& key) const;

HashMap NULL … private: struct node { Key key; Value value; node *next; }; node **buckets; int numBuckets; int count; int hash(const Key& key) const; // Q: Can you draw the HashMap // object in this memory diagram, // including filling in values for // all fields? 1 2 3 4 5 6 7

Hash key collisions & Big-O of HashMap If there are no collisions, find/add/remove are all O(1)—just compute the key and go! Two factors for ruining this magical land of instantaneous lookup: Too-small table (worst case = 1) Hash function doesn’t produce a good spread int awfulHashFunction(string input) { return 4; } Find/add/remove all O(n) worst case // h/t http://xkcd.com/221/

The Birthday Problem Commonly mentioned fact of probability theory: In a group of n people, what are the odds that 2 people have the same birthday? (assume birthdays are uniformly distributed across 365 days, which is wrong in a few ways) For n ≥ 23, it is more likely than not For n ≥ 57, >99% chance Moral of the story: hash tables almost certainly have at least one collision